Chapter 3: Q16E (page 117)
Find the quantile function for the distribution in Example 3.3.1.
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Two students,AandB,are both registered for a certain course. Assume that studentAattends class 80 percent of the time, studentBattends class 60 percent of the time, and the absences of the two students are independent. Consider the conditions of Exercise 7 of Sec. 2.2 again. If exactly one of the two students,AandB,is in class on a given day, what is the probability that it isA?
Question:Suppose that in a certain drug the concentration of aparticular chemical is a random variable with a continuousdistribution for which the p.d.f.gis as follows:
\({\bf{g}}\left( {\bf{x}} \right){\bf{ = }}\left\{ \begin{array}{l}\frac{{\bf{3}}}{{\bf{8}}}{{\bf{x}}^{\bf{2}}}\;{\bf{for}}\;{\bf{0}} \le {\bf{x}} \le {\bf{2}}\\{\bf{0}}\;{\bf{otherwise}}\end{array} \right.\)
Suppose that the concentrationsXandYof the chemicalin two separate batches of the drug are independent randomvariables for each of which the p.d.f. isg. Determine
(a) the joint p.d.f.of X andY;
(b) Pr(X=Y);
(c) Pr(X >Y );
(d) Pr(X+Y≤1).
Suppose that three random variables X1, X2, and X3 have a continuous joint distribution with the following joint p.d.f.:
\({\bf{f}}\left( {{{\bf{x}}_{\bf{1}}}{\bf{,}}{{\bf{x}}_{\bf{2}}}{\bf{,}}{{\bf{x}}_{\bf{3}}}} \right){\bf{ = }}\left\{ {\begin{align}{}{{\bf{c}}\left( {{{\bf{x}}_{\bf{1}}}{\bf{ + 2}}{{\bf{x}}_{\bf{2}}}{\bf{ + 3}}{{\bf{x}}_{\bf{3}}}} \right)}&{{\bf{for0}} \le {{\bf{x}}_{\bf{i}}} \le {\bf{1}}\,\,\left( {{\bf{i = 1,2,3}}} \right)}\\{\bf{0}}&{{\bf{otherwise}}{\bf{.}}}\end{align}} \right.\)
Determine\(\left( {\bf{a}} \right)\)the value of the constant c;
\(\left( {\bf{b}} \right)\)the marginal joint p.d.f. of\({{\bf{X}}_{\bf{1}}}\)and\({{\bf{X}}_{\bf{3}}}\); and
\(\left( {\bf{c}} \right)\)\({\bf{Pr}}\left( {{{\bf{X}}_{\bf{3}}}{\bf{ < }}\frac{{\bf{1}}}{{\bf{2}}}\left| {{{\bf{X}}_{\bf{1}}}{\bf{ = }}\frac{{\bf{1}}}{{\bf{4}}}{\bf{,}}{{\bf{X}}_{\bf{2}}}{\bf{ = }}\frac{{\bf{3}}}{{\bf{4}}}} \right.} \right){\bf{.}}\)
In Example 3.8.4, the p.d.f. of \({\bf{Y = }}{{\bf{X}}^{\bf{2}}}\) is much larger for values of y near 0 than for values of y near 1 despite the fact that the p.d.f. of X is flat. Give an intuitive reason why this occurs in this example.
Suppose that\({X_1}....{X_n}\)are i.i.d. random variables, each having the following c.d.f.:\(F\left( x \right) = \left\{ \begin{array}{l}0\,\,\,\,\,\,\,\,\,\,\,\,\,\,for\,x \le 0\\1 - {e^{ - x}}\,\,\,for\,x > 0\end{array} \right.\)
Let\({Y_1} = min\left\{ {{X_1},{X_2}..{X_n}} \right\}\)and\({Y_n} = max\left\{ {{X_{1,}}{X_2}..{X_n}} \right\}\)Determine the conditional p.d.f. of\({Y_1}\)given that\({Y_n} = {y_n}\)
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